Use the generalised mean value theorem to prove this

In summary, the question is asking to prove the generalized mean value theorem using given hints and the functions F(x) and G(x). The generalized mean value theorem states that for continuous and differentiable functions F and G, there exists a value c in the interval (a, b) where F'(c)/G'(c) is equal to the difference of F(b)-F(a) over G(b)-G(a). To solve this, F and G are substituted into the mean value theorem equation and algebraically manipulated to arrive at the given equation.
  • #1
chipotleaway
174
0

Homework Statement


Let f(x) be a continuous function on [a, b] and differentiable on (a, b). Using the generalised mean value theorem, prove that:

[tex]f(x)=f(c) + (x-c)f'(c)+\frac{(x-c)^2}{2}f''(\theta)[/tex] for some [tex]\theta \in (c, x)[/tex]


Homework Equations


Hints given suggest consdiering F(x) = f(x) - f(c) - f'(c)(x-c) and G(x) = (x-a)^2
(F and G and their derivatives are the functions that appear in the given mean value theorem)


The Attempt at a Solution


I don't have no clue as to how to proceed other than making the substitutions in the hints and plugging it into the MVT equation and hoping the function I'm after pops out but this doesn't seem right - especially for the number of marks it's worth (plus I'm not really getting anywhere with it anyway).
 
Physics news on Phys.org
  • #2
What is the generalized mean value theorem?
 
  • #3
Cauchy's mean value theorem. If F and G are continuous on [a, b] and differentiable on (a, b) and G'(x)≠0 for all x in (a, b) then there exists some c in (a, b) such that:

[tex]\frac{F'(c)}{G'(c)}=\frac{F(b)-F(a)}{G(b)-G(a)}[/tex]
 
  • #4
chipotleaway said:
Cauchy's mean value theorem. If F and G are continuous on [a, b] and differentiable on (a, b) and G'(x)≠0 for all x in (a, b) then there exists some c in (a, b) such that:

[tex]\frac{F'(c)}{G'(c)}=\frac{F(b)-F(a)}{G(b)-G(a)}[/tex]

OK, then you should just plug in your functions ##F(x)=f(x)-f(x)-f^\prime(c)(x-c)## and ##G(x) = (x-c)^2## into this mean-value theorem.

Note however, that the numbers ##c##, ##a## and ##b## in this mean-value theorem are not the same as the ones in the OP!
 
  • #5
Thanks! I'll have to check my algebra then, haha.
 
  • #6
If [itex]F(x)=f(x)-f(c)-f'(c)(x-c)[/itex], then when I differentiate, do I treat f(c) and f'(c) as constants? I'm getting [itex]F'(x)=f'(x)-f''(c)(x-c)+f'(c)[/itex]
 
  • #7
chipotleaway said:
If [itex]F(x)=f(x)-f(c)-f'(c)(x-c)[/itex], then when I differentiate, do I treat f(c) and f'(c) as constants?

Yes. They are constants.
 

Related to Use the generalised mean value theorem to prove this

What is the generalised mean value theorem?

The generalised mean value theorem is a fundamental theorem in calculus that states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point within the interval where the slope of the tangent line is equal to the average rate of change of the function over the interval.

What is the importance of the generalised mean value theorem?

The generalised mean value theorem is important because it allows us to prove the existence of a certain point within an interval where the function has a specific rate of change. This theorem is also used to prove other important theorems in calculus, such as the fundamental theorem of calculus.

How do you use the generalised mean value theorem to prove a statement?

To use the generalised mean value theorem to prove a statement, you must first identify the function and the closed interval on which it is continuous and differentiable. Then, you must show that the function satisfies the conditions of the theorem, and use the theorem to find the specific point within the interval that satisfies the statement.

Can the generalised mean value theorem be used for any type of function?

No, the generalised mean value theorem can only be used for continuous and differentiable functions. This means that the function must be defined and have a slope at every point within the closed interval.

What are the applications of the generalised mean value theorem?

The generalised mean value theorem has various applications in mathematics, including calculating error bounds in numerical analysis, proving the existence of solutions to differential equations, and finding critical points in optimization problems.

Similar threads

Mean Value Theorem: Proof & Claim
    • Calculus and Beyond Homework Help
Replies
11
Views
1K
Intermediate Value Theorem Problem on a String
    • Calculus and Beyond Homework Help
Replies
14
Views
1K
Attempting to prove the Intermediate Value Theorem
    • Calculus and Beyond Homework Help
Replies
26
Views
2K
Calculate this Integral around the Circular Path using Green's Theorem
    • Calculus and Beyond Homework Help
Replies
3
Views
351
Find domain where function is Lipschitz
    • Calculus and Beyond Homework Help
Replies
2
Views
691
Linearly independent functions with identically zero Wronskian
    • Calculus and Beyond Homework Help
Replies
1
Views
340
Lagrange multipliers understanding
    • Calculus and Beyond Homework Help
Replies
8
Views
510
Using the mean value theorem to prove the chain rule
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Verifying Hypotheses of the Mean Value Theorem for f(x)=1/(x-2) on [1,4]
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
I Continuity of Mean Value Theorem
    • Calculus
Replies
12
Views
572

Hot Threads

Recent Insights

Back
Top